Math 2280-001
Fri Apr 10
7.1-7.2 Laplace transform, and application to DE IVPs
, The Laplace transform is a linear transformation "L" that converts piecewise continuous functions
f t , defined for t R 0 and with at most exponential growth ( f t % CeM t for some values of C and
M), into functions F s defined by the transformation formula
N
F s =L f t
f t eKs t dt .
s d
0
, Notice that the integral formula for F s is only defined for sufficiently large s, and certainly for
s O M, because as soon as s O M the integrand is decaying exponentially, so the improper integral from t
= 0 to N converges.
, The convention is to use lower case letters for the input functions and (the same) capital letters for their
Laplace transforms, as we did for f t and F s above. Thus if we called the input function x t then we
would denote the Laplace transform by X s .
Taking Laplace transforms seems like a strange thing to do. And yet, the Laplace transform L is just one
example of a collection of useful "integral transforms". L is especially good for solving IVPs for linear
DEs, as we shall see starting today. Other famous transforms - e.g. Fourier series and Fourier transform
are extremely important in studying linear differential and partial differential equations. We will discuss
Fourier series in about a week. These transforms are also studied in Math 3140, 3150, and in various
5000-level pure and applied math classes.
Exercise 1) Use the definition of Laplace transform
N
F s =L f t
f t eKs t dt
s d
0
to check the following facts, which you will also find inside the front cover of your text book.
1
a) L 1 s =
sO0
s
1
b) L ea t s =
(s O a if a 2 =, s O a if a = a C k i 2 C )
sKa
c) Laplace transform is linear, i.e.
L f1 t C f2 t s = F1 s C F2 s .
L cf t s =cF s .
d) Use linearity and your work above to compute L 3 K 4 eK2 t s .
For the linear differential equations and systems of differential equations that we've been studying, the
following Laplace transforms are very important:
Exercise 2) Use complex number algebra, including Euler's formula, linearity, and the result from 1b that
1
L e aCki t s =
sK aCk i
to verify that
s
a) L cos k t s = 2
s C k2
k
b) L sin k t s = 2
s C k2
sKa
c) L ea t cos k t
s =
s K a 2 C k2
k
d) L ea t sin k t s =
.
s K a 2 C k2
(Notice that if we tried doing these Laplace transforms directly from the definition, the integrals would be
messy but we could attack them via integration by parts or integral tables.)
It's a theorem (hard to prove but true) that a given Laplace transform F s can arise from at most one
piecewise continuous function f t . (Well, except that the values of f at the points of discontinuity can be
arbitrary, as they don't affect the integral used to compute F s . ) Therefore you can read Laplace
transform tables in either direction, i.e. not only to deduce Laplace transforms, but inverse Laplace
transforms LK1 F s
t = f t as well.
Exercise 3) Use the Laplace transforms we've computed and linearity to compute
7
1
10 s
LK1
C 2
K 2
t .
s
s C 16
s C 16
f t
f t % CeM t
c f t Cc f t
1 1
2 2
f t eKs t dt
0
for s O M
c F s Cc F s
1 1
2 2
1
s
(s O 0
1
ea t
N
Fs d
1
(s O R a
sKa
s
cos k t
(s O 0
s2 C k2
sin k t
k
(s O 0
s2 C k2
ea tcos k t
sKa
2 C k2
sKa
ea tsin k t
(s O a
k
sKa
sOa
2 C k2
f# t
s F s Kf 0
f ## t
s2F s K s f 0 K f # 0
Laplace transform table
The integral transforms of DE's and PDE's were designed to have the property that they convert the
corresponding linear DE and PDE problems into algebra problems. For the Laplace transform it's because
of these facts:
Exercise 4a) Use integration by parts and the definition of Laplace transform to show that
L f# t
s =sL f t
s Kf 0 = s F s Kf 0 .
4b) Use the result of a, applied to the function f# t to show that
L f## t s = s2 F s K s f 0 K f# 0 .
4c) What would you guess is the Laplace transform of f### t ? Could you check this?
Here's an example of using Laplace transforms to solve DE IVPs, in the context of Chapter 3 and the
mechanical (and electrical) application problems we considered there.
Exercise 5) Consider the undamped forced oscillation IVP
x## t C 4 x t = 10 cos 3 t
x 0 =2
x# 0 = 1
If x t is the solution, then both sides of the DE are equal. Thus the Laplace transforms are equal as well...
.so, equate the Laplace transforms of each side and use algebra to findL x t s = X s . Notice you've
computed X s without actually knowing x t ! If you were happy to stay in "Laplace land" you'd be
done. In any case, at this point you can use our table entries to find x t = LK1 x t s .
(Notice that if your algebra skills are good you've avoided having to use the Chapter 3 algorithm of (i) find
xH (ii) find an xP (iii) x = xP C xH (iv) solve IVP.) Magic! Or, would you have prefered to convert to a
first order system and to have used variation of parameters with an FM or eA t , like in Chapter 5 :-) ? )
> with DEtools :
> dsolve x## t C 4$x t = 10$cos 3$t , x 0 = 2, x# 0 = 1
; # to check
Exercise 6) Use Laplace transform as above, to solve the IVP for the following underdamped, unforced
oscillator DE:
x## t C 6 x# t C 34 x t = 0
x 0 =3
x# 0 = 1
> dsolve
>
x## t C 6$x# t C 34$x t = 0, x 0 = 3, x# 0 = 1
; # to check